We analyse the structure of enveloping surface in (1+1)D and in (2+1)D models of ballistic growth and calculate the distribution function of number of maximal points (i.e., local “peaks”) of such a surface. Our computation uses two facts: (a) the uniform one-dimensional ballistic growth process in the steady state can be formulated in terms of “rise-and-descent” patterns in the ensemble of random permutation matrices, and (b) the statistics of “rises” and “descents” in random permutations can be described in terms of a certain continuous-space Hammersley-type process. Besides, a short review of related problems is given: (i) the relation between the statistics of random ballistic growth and statistics of entanglements in randomly growing braids is discussed; (ii) the exact asymptotic results for the statistics of asymmetric (1+1)D ballistic deposition by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers are presented.

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